\chapter{Applications to differential equations}


Of course, the main propsect of Lie theory is its use in determining smooth transformation groups for differential equations!



\section{Prolongations}

To construct a framework in which we can apply Lie theory, we consider the space solutions of a differential equation as a subvariety of a jet space; this is our manifold.
Given a differential equation $f(x,u^{(n)})=0$, we desire to find a smooth transformation group $G$; a Lie group with a group action on the jet space that is $G$-invariant over $f$.






\subsection{Prolongation of IGs}

Since the Lie group is the object to be calculated, we must consider a way to prolong infinitesimal generators, corresponding to the prolongation of the group action.


\begin{theorem}
	Let $G$ be a Lie group and $f(x,u^{(n)})$ be a differential equation of maximal rank, and for any element $\mathbf{V} \in \mathfrak{g}$
\[\mathrm{Pr} \mathbf{V}[ f(x,u^{(n)})] =0\]
Then $G$ is a smooth transformation group for the differential equation
\end{theorem}

\subsection{Prolongation formula}



This is the key to determining the Lie algebra, from which the exponential map can recover the Lie group.


\subsection{Properties of prolonged IGs}
is linear, can pass within Lie bracket